
The multigroup low-order QD (LOQD) equations for the angular moments given by
\(\frac{\partial E_g }{\partial t} + \frac{\partial F_g }{\partial x} +c\varkappa _{g}(T) E_g = 4\pi \varkappa _{g}(T) B_g(T) \, ,\\\frac{1}{c} \frac{\partial F_g}{\partial t} +c \frac{\partial }{\partial x} \Big ( f_g[I] E_g \Big ) + \varkappa _{g} (T) F_g= 0 \, ,\) 

where
\(E_g = \frac{1}{c} \int _{-1}^1 I_g d \mu \) 
is the group radiation energy density,
\(F_g = \int _{-1}^1 \mu I_g d \mu \)  is the group radiation flux, and

\( f_g\big [ I \big ] = \left.\int _{-1}^1 \mu ^2 I_g d \mu \right\bad. \int _{-1}^1 I_g d \mu \) 
is the group QD (Eddington) factor that provides closure of the BTE and multigroup LOQD equations.
This closure is exact when \(f_g\)  is defined by the solution of the BTE (REF ) according to the QD (VEF) method [1]}, [2]}.

The effective grey LOQD equations for
the total radiation energy density \(E= \sum _{g=1}^{N_g} E_g\)  and
the total flux \(F= \sum _{g=1}^{N_g} F_g\)  are given by
\(\frac{\partial E }{\partial t}+ \frac{\partial F }{\partial x}+ c \bar{\varkappa }_E E= c \bar{\varkappa }_B a_R T^4 \, ,\\\frac{1}{c}\frac{\partial F }{\partial t}+ c\frac{\partial (\bar{f} \big [I \big ] E) }{\partial x}+ \bar{\varkappa }_R F + \bar{\eta }E = 0 \, , \) 

where the grey coefficients are
\(\bar{\varkappa }_{E}= \frac{\sum _{g=1}^{N_g} \varkappa _{g}E_g}{\sum _{g=1}^{N_g} E_g} \, ,\quad \bar{\varkappa }_{B}= \frac{\sum _{g=1}^{N_g} \varkappa _{g} B_g}{\sum _{g=1}^{N_g} B_g} \, ,\quad \bar{\varkappa }_{R}= \frac{\sum _{g=1}^{N_g} \varkappa _{g}|F_g|}{\sum _{g=1}^{N_g} |F_g|} \ ,\\\bar{f} = \frac{\sum _{g=1}^{N_g} f_g E_g}{\sum _{g=1}^{N_g} E_g} \, ,\quad \bar{\eta }= \frac{\sum _{g=1}^{N_g} (\varkappa _{g} - \bar{\varkappa }_{R}) F_g}{\sum _{g=1}^{N_g} E_g} \, .\) 

